Delay-Throughput Performance in Mobile Ad-HocNetworks with Heterogeneous Nodes

Valentina Martina ∗ Michele Garetto † Emilio Leonardi ∗

† Dipartimento di Informatica, Università di Torino, Italy∗ Dipartimento di Elettronica, Politecnico di Torino, Italy

ABSTRACTIn this paper, we analyze asymptotic delay-throughput performanceof mobile ad-hoc networks comprising heterogeneous nodes withrestricted mobility. In particular, we consider a scenario in whicheach node moves around one or more home-points (in a finite num-ber) randomly placed over the area. For such restricted mobilitymodel, we propose a new class of scheduling and routing schemes,which significantly outperforms all delay-throughput results previ-ously obtained.

1. INTRODUCTIONFundamental scaling laws 1 of ad-hoc wireless networks have beenextensively studied in the past several years, starting from the sem-inal works of Gupta-Kumar [1] and Grossglauser-Tse [2]. In theusual setting, Θ(n) nodes are placed over a finite region of theplane and randomly establish connections among them formingΘ(n) source-destination (S-D) pairs. Sources can employ multi-hop communications to reach their destinations, using other nodesas relays. Concurrent transmissions on the same wireless channelare subject to mutual interference, which limits the overall systemperformance.

In the case of mobile nodes, the novel store-carry-forward com-munication scheme, according to which nodes can physically carrydata stored in their buffer before forwarding them to the next hop,has recently attracted lots of attention due to the increasing popu-larity of so called delay-tolerant networks [3] based on human [4]or vehicular [5] mobility. In this scenario, asymptotic analysisof capacity and delay as the number of nodes increases have be-ing mainly carried out under the assumption that nodes are iden-tical and uniformly visit the entire network area. Several mobilitymodels have been analyzed, ranging from the simple re-shufflingmodel [6, 7], to the Brownian motion [8], and variants of random

9 1Given two functions f(n) ≥ 0 and g(n) ≥ 0: f(n)= o(g(n))means limn→∞ f(n)/g(n) = 0; f(n) = O(g(n)) meanslim supn→∞ f(n)/g(n) = c < ∞; f(n) = ω(g(n)) is equiv-alent to g(n) = o(f(n)); f(n) = Ω(g(n)) is equivalent to g(n) =O(f(n)); f(n) = Θ(g(n)) means f(n) = O(g(n)) and g(n) =O(f(n)); at last f(n) ³ g(n) means limn→∞ f(n)/g(n) = 1.

walk and random way-point [9,10]. All of the above mobility mod-els are rather generous, since they allow the trajectory of each nodeto ‘fill the space’ over time, uniformly covering the entire networkarea. Such ‘homogeneous mixing’ assumption on the nodes’ mo-bility process has been shown to provide optimal performance interms of network capacity alone [2]. Whether this same assump-tion is also optimal in terms of delay-throughput trade-offs is aninteresting question that we address in this paper, showing that uni-form mobility can be largely suboptimal when both capacity anddelay are considered.

Indeed, both every-day life experience and measurement data col-lected from real mobility traces [11, 12] suggest that the mobilitypattern of an individual node is rather restricted over the networkarea, as users spend most of the time in proximity of a few preferredplaces, and rarely go outside their region of habit.

To the best of our knowledge, only few works have analyzed theimpact of restricted mobility models on asymptotic throughput anddelay. In [16] the authors consider a one-dimensional mobilitymodel in which each node uniformly visits a randomly chosengreat circle on the unit sphere, and obtain that a constant per-nodethroughput is achievable by each node, just like in the originalGrossglauser-Tse scenario [2]. The work in [17] considers the samerestricted mobility model, showing that also the delay is the sameas in the case of uniform mobility over the entire network area.

In [18, 19] we have considered a two-dimensional, restricted mo-bility model which produces, for each node, a rotationally invari-ant stationary distribution centered at a home-point uniformly cho-sen in the area; the resulting throughput varies with continuityin between the two extreme cases of static nodes (Gupta-Kumar)and fully mobile nodes (Grossglauser-Tse), depending on how thephysical network extension scales with respect to the average dis-tance reached by the nodes from their home-point. This result con-firms that throughput alone is maximized when the nodes span theentire network area.

In this work we use the same restricted mobility model introducedin [18,19], extending the analysis to delay-throughput performance.To this aim we will adopt the simple re-shuffling model (also re-ferred to as i.i.d. mobility model) already considered in [6, 7] inthe case of nodes uniformly visiting the network space. Our mainfinding is that restricted mobility can be very beneficial in terms ofthroughput and delay, as it allows to achieve superior performance(in scaling order) than in the case of homogeneous node mixing.This result suggests that realistic delay-tolerant networks, whichare often characterized by restricted mobility patterns, can scale

much better that what previously believed.

2. SYSTEM ASSUMPTIONS2.1 Mobility modelWe consider an extended network composed of n nodes movingover a square regionO of area n with wrap-around conditions (i.e.,a torus), to avoid border effects. Note that, under this assumption,the node density over the area remains constant and equal to 1 aswe increase n. Time is divided into slots of equal duration, whichis normalized to 1.

In our work, we account for the restricted mobility pattern of nodesover the network area as follows. We assume that each node movesaround one or more ‘home-points’ (in a finite number) randomlyplaced over the area. As an example, one home-point could bethe residence of a person, and another home-point could be his/herworkplace. At any given time we say that a node is associatedto one of its home-points, meaning that it is currently moving inproximity of it. The node periodically migrates from one of itshome-point to another. The exact details about how these migra-tions occur are not important. We simply assume that a node, whilebeing associated to any of its home-points, spends on average a fi-nite number of slots before migrating to another home-point. Wefurther assume that the time taken to switch from one home-pointto another is negligible (or at least uniformly bounded), and that anode does not have the opportunity of exchanging data with othernodes while traveling from one home-point to another (the contri-bution of such data exchanges to the asymptotic network capacityis, in any case, not important in order sense).

As a consequence of the above assumptions, each node spends afinite fraction of time in proximity of each of its home-points, andmigrates in finite time from one home-point to another. In or-der to compute the resulting scaling laws of throughput and de-lay, such mobility process can be equivalently described by a two-dimensional i.i.d. mobility model, in which the positions of thenodes are totally reshuffled after each slot, independently from slotto slot and among the nodes. At the beginning of each slot, a nodejumps in zero time to a new position randomly selected accordingto a given spatial stationary distribution over the network area.

In our model, the overall spatial stationary distribution of a nodecan be obtained as follows. We first define the spatial stationarydistribution of a node conditioned to the fact that it is currentlyassociated to a given home-point. The overall spatial distributionis then given by a mixture of the spatial distributions of the nodearound its home-points, each of them weighted by the (positive)probability that the node is associated to the corresponding home-point. It follows that we only need to specify: i) the stationarydistribution of a node around each home-point; ii) the distributionover the network area of the home-points of a node.

Spatial distribution around each home-point. The spatial sta-tionary distribution of a node around each of its home-points isassumed to be rotationally invariant with respect to the home-point, and thus can be described by a generic, non increasing func-tion φ(d) of the distance2 d from the home-point. In this paperwe assume that φ(d) decays as a power-law of exponent δ, i.e.,

9 2Given any two points X1 = (x1, y1)∈O andX2 = (x2, y2)∈O we define their distance over the torussurface as:‖X1 −X2‖ = min

u,v∈±√n,0

√(x1 + u− x2)2 + (y1 + v − y2)2

φ(d) ∼ d−δ . This choice is supported by a number of measure-ments papers which have found power-laws to be quite ubiquitousin experimental traces related to both human and vehicular mobil-ity [12–15].

To avoid divergence problems in proximity of the home-point, wetake function s(d) = min(1, d−δ), and normalize it so as to obtaina proper probability density function over the network area:

φ(d) =s(d)∫∫O s(d)

The value of the normalization constant G(δ) =∫∫O s(d) can be

approximated, in order sense, by the following integral in polarcoordinates:

G(δ) = Θ

(∫ 2π

0

dθ

∫ √n

0

min(1, ρ−δ)ρ dρ

)

We obtain that G(δ) is finite for any δ > 2. For 0 ≤ δ < 2

we have G(δ) = Θ(n2−δ2 ). For the special value δ = 2 we have

G(δ) = Θ(log n). Note that δ = 0 leads to a uniform distributionover the space, whereas letting δ go to infinity we obtain the samebehavior as that of a static network.

Spatial distribution of the home-points of a node. The spatialdistribution of the home-points of a node is modelled as follows.One home-point, which is called the primary home-point of thenode, is placed uniformly at random in the network area. Then,in the case of multiple home-points, all of the other home-pointsare randomly placed according to a rotationally invariant distribu-tion centered at the primary home-point, which can be completelyspecified by a function ψ(d) analogous to the one introduced aboveto describe the stationary distribution of a node around one home-point, i.e., analogous to function φ(d) previously defined. Again,we assume that ψ(d) decays as a power-law, with a different expo-nent ζ, i.e., ψ(d) ∼ d−ζ , with ζ < δ. We assume ζ < δ becausewe want the home-points of a node to be well distinct from eachother, and thus separated by an average distance larger than thetypical distance reached by a node from one home-point. Other-wise, when ζ ≥ δ, the network performance would be the same asin the case of a single home-point per node.

Since the number of home-points of a node is assumed to be uni-formly bounded by a constant, it happens that all the performancegain (in terms of network capacity and delay) resulting from havingmultiple home-points per node can be already achieved in the caseof just two home-points per node. For this reason, in the followingwe will restrict our analysis to the case in which each node i has ei-ther a single home-point H1

i , or two home-points Hi = H1i , H2

i .

Let Xi(t) denote the position of node i at time t (t isan integer denoting the slot sequence number) and X(t) =(X1(t), X2(t) . . . Xn(t)) be the vector of nodes’ positions at timet; we define by dij(t) the distance between mobile i and mobile jat time t, i.e., dij(t) = ‖Xi(t)−Xj(t)‖

We define by dHij the minimum distance between the home-points

of nodes i and j. In the case of a single home-point per node,dH

ij = ‖Hi − Hj‖. In the case of two home-points per node, wehave

dHij = min

h,h′=1,2

‖Hh

i −Hh′j ‖

(1)

One final assumption concerns the time-scale of node mobility withrespect to the packet transmission time. In previous work dealingwith asymptotic delay-throughput trade-offs, two different time-scales have been analyzed: i) the fast-mobility case, in which onlysingle-hop transmissions are allowed during a slot, being the packettransmission time comparable to the duration of a slot; ii) the slow-mobility case, in which multi-hop communications can occur dur-ing a slot, by properly scaling down the packet transmission time.In this work we will focus on the more restrictive fast-mobility case,in which the packet transmission time matches the duration of aslot.

2.2 Communication modelTo account for interference among simultaneous transmissions, weadopt the protocol model introduced in [1] 3. Nodes employ a com-mon range R for all transmissions which occur in the same timeslot. A transmission from node i to node j using transmission rangeR can be successfully received at node j if and only if the followingtwo conditions hold:

1. the distance between i and j is smaller than or equal to R,i.e., dij(t) ≤ R

2. for every other node k simultaneously transmitting,dkj(t) ≥ (1 + ∆) R, being ∆ a guard factor.

Transmissions occur at fixed rate which is normalized to 1. Weassume that a single copy of each packet is present in the networkat any time, i.e., data units are not broadcasted nor replicated, andnodes do not keep copies of previously received packets in theirbuffer.

2.3 Traffic modelSimilarly to previous work we consider permutation traffic patternsin which n randomly selected source-destination pairs exchangetraffic at rate λ. Source-destination pairs are selected is such a waythat every node is origin and destination of a single traffic flow withaverage rate λ. We further assume that messages are generatedat every source according to independent memoryless Bernoullianprocesses.

2.4 Throughput and delayWe use the following definitions of asymptotic throughput and de-lay. Let Li(T ) be the number of packets delivered to the destinationof node i in the time interval [0, T ]. The delay of a packet is thetime it takes to the packet to reach the destination after it is gener-ated at the source. Let Di(t) be the sum of the delays experiencedby all packets successfully delivered to the destination of node iin the time interval [0, T ]. We say that an asymptotic throughputλ and an asymptotic delay D per S-D pair are feasible if there isan n0 such that for any n ≥ n0 there exists a scheduling/routingscheme for which both the following properties hold

limT→∞

Pr

(Li[T ]

T≥ λ, ∀i

)= 1

limT→∞

Pr

(Di[T ]

Li[T ]≤ D, ∀i

)= 1

9 3The protocol model has been proven to be pessimistic with re-spect to the physical model employing power control (see Theorem4.1, pag. 174 in [20]). Thus the results obtained in this paper canbe regarded as lower bounds of the network performance achiev-able under the physical model employing power control.

Equivalently, we say in this case that the network sustains an ag-gregate throughput Λ = nλ. We will also adopt the simple PowerFunction [21], defined as the ratio λ/D, to characterize the systemperformance by a single metric.

3. PREVIOUS RESULTS FOR THE I.I.D.FAST MOBILITY MODEL

To avoid confusion, we limit ourselves to reporting existing scal-ing laws obtained for the i.i.d. mobility model, in the case of fastmobiles.

In [6] the authors analyze throughput/delay trade-offs in the case ofnodes uniformly visiting the entire network area, with or withoutpacket replication. The following general trade-off is established:

λ = O

(D

n

)(2)

In particular the two-hop scheme of Grossglauser-Tse is proven toincur a delay D = Θ(n) while guaranteeing a per-node through-put λ = Θ(1). To improve delay, two different schemes exploit-ing packet redundancy are proposed: the first is still based ontwo hops, and achieves D = Θ(

√n) and λ = Θ(1/

√n); the

second employs multiple hops and achieves better delay perfor-mance D = O(log n) while sacrificing the per-node throughputλ = O (1/(n log n)).

It has been shown in [7] that previous results can be further im-proved by encoding transmitted information at sources. A classof joint coding-scheduling-routing schemes is introduced whichachieves a trade-off

λ = Θ

(√D

n

)(3)

in the case of i.i.d. mobility with fast mobiles, when D is bothω( 3√

n) and o(n).

In [22] we have already analyzed the delay-throughput trade-offsachievable under the restricted mobility model considered here,limiting our attention to the case of a single home-point per node.However, in [22] we have proposed a class of scheduling-routingschemes specifically tailored to the case of δ < 2, i.e., for nodeswith broad mobility over the network area. Such schemes turnsout to be largely suboptimal when δ > 2, i.e., for more restrictedmobility patterns. With respect to [22], in this paper we pro-vide the following contributions: i) we propose a different classof scheduling-routing schemes, which are tailored to the case ofδ > 2, and achieve much better performance than that reportedin [22]; ii) we were able to prove that our scheme jointly achievesnear-optimal throughput and delay performance for 2 ≤ δ < 3;iii) we analyze the case of multiple home-points per node, show-ing how the network performance can be improved by exploitingrestricted mobility patterns around different home-points.

4. ALGORITHM DESCRIPTION4.1 The routing schemeWe propose a routing scheme according to which messages ad-vance along a chain of relay nodes whose home-points becomeprogressively closer (in terms of the minimal home-point distancespecified by (1)) to the home-points of the destination.

More in detail, we recursively partition the network area intosquarelets of smaller and smaller size. At the first iteration the do-

a

d

b

Figure 1: Example of routing of a message generated by nodea in routing step 0. Dots denote home-points of nodes.

main O is divided into four disjoint squares of edge√

n/2. Ateach iteration i (i ≤ imax) every square obtained in the previousiteration is in turn divided into four disjoint squarelets. By so do-ing, i-level squarelets have edge zi =

√n/2i, and their cardinality

is 4i. We denote by Sik, for 1 ≤ k ≤ 4i, each i-level squarelet,

0 ≤ i ≤ imax. We say that two i-level squares are adjacent ifthey are originated from the same (i − 1)-level square. Noticethat S0

1 = O. Parameter imax determines the edge of the smallestsquarelet, zimax =

√n/2imax . At last, we denote by |Si| = z2

i thearea of a level-i squarelet.

We say that two nodes a and b are i-level neighbors if the smallestsquarelet containing both a home-point of a and a home-point of bis a i-level squarelet. Suppose that node a has to forward a packetm directed to a destination d which is a i-level neighbor of a. Inthis case, while being handled by a, we say that m is in step i ofthe routing algorithm.

If i < imax, node a sends the packet to a node b which is an (i+1)-level neighbor of d. If i = imax, node a sends the packet directlyto its final destination d. By construction, packets go through atmost imax intermediate relay nodes before being delivered to thedestination. Figure 1 illustrates an example in which imax = 3,and a message is generated by node a in routing step 0, destined tonode d. Here the message has to go through 3 relay nodes beforearriving in a squarelet of minimum size containing the home-pointof the destination.

To guarantee the existence, uniformly in the network, of nodes bwhich are (i + 1)-level neighbors of d, we need to have |Si| =Ω(log n) for any i ≤ imax. This comes from a standard concen-tration result of the Poisson point process widely used in relatedwork, and not repeated here. As a consequence, the smallest pos-sible edge is zimax = Ω(

√log n), where imax = 1/2 log2 n −

log2 zimax ≤ 2 log2 n − log2 log n = Θ(log n). In the followingwe will better specify how to properly select zimax .

4.2 The scheduling schemeA scheduling scheme is in charge of selecting, at each slot, the setof (non-interfering) transmitter-receiver pairs to be enabled in thenetwork, as well as the message to be transmitted over each enabledpair. In our family of schemes, the transmission range employed bya transmitter depends on the routing step reached by the messageto be sent. To better pack simultaneous transmissions, and thusmaximize the network throughput, our scheduling policy selects,

Ai

Figure 2: Example of non-interfering subset of cells with M =9.

in a given slot, transmissions having homogeneous transmissionranges. This is done by selecting, at the beginning of each slot, astep i according to an assigned probability distribution ps

i . Then theslot is reserved only to messages which are in step i of the routingalgorithm, i.e., to messages currently stored at nodes which are i-level neighbors of the destination.

A common transmission range Ri = O(√

n

2i

)is employed by all

communications occurring during a slot devoted to step i of therouting scheme. Once step i has been selected, every squarelet Si

k

is divided into square cellsAi of area Ai = R2i = O(z2

i ), forminga regular square tessellation.

According to the protocol model, at most one transmission can beenabled in each cell. Moreover, one can easily construct M =Θ(1) subsets of regularly spaced, non-interfering cells (for exam-ple, M = 9 assuming a protocol model with ∆ = 0, see Figure2). Each subset can then be enabled in one out of M slots, guaran-teeing fairness among all cells and absence of interference amongconcurrent transmissions.

Within a given cell Ai, the specific transmitter-receiver pair to beenabled is selected as follows. First a node a currently staying inAi and such that: i) a has a home-point in Ai; ii) a is storing amessage m in routing step i, is selected at random as transmitter,if there is any. Let d be the final destination of message m. Ifi = imax, node d is the only possible receiver, hence it must befound inAi for the communication to occur. Instead, for i < imax,a node b (if there is any) is randomly selected as receiver amongthe (i + 1)-level neighbors of d currently staying in Ai. Note thatby construction home-points of a and b lie in adjacent (i+1)-levelsquares.

If such a transmitter-receiver pair is found, cell Ai is declared asactive and the transmission from a to b (d in the case of i = imax)takes place.

We further assume that messages m in routing step i ≤ imax areselected by node a according to a FIFO scheduling policy. Thus,logically each node is equipped with a FIFO queue for each routingstep (Figure 3).

5. PERFORMANCE EVALUATION5.1 Design considerations

Given that step i has been selected at the beginning of a slot, byconstruction the number Ni of parallel transmissions that can occurin the network during the slot equals the number of active cells. Onaverage we have:

E[Ni] =n

MAiP(active cell | i) (4)

where n is the total network area, M = Θ(1) is a constant account-ing for interference, and P(active cell | i) is the probability that ageneric cell is active at step i.

Note that, on the one hand, areas Ai should be minimized so as tomaximize the spatial reuse; on the other hand, if areas Ai are settoo small then P(active cell | i) becomes arbitrarily small as well,vanishing the potential advantage of increasing the spatial reuse.A good design choice is to minimize Ai while guaranteeing thatlimn→∞ P(active cell | i) > 0.

Unfortunately a direct evaluation of P(active cell | i) is rather dif-ficult, because P(active cell | i) depends on both node positionsand traffic conditions. Thus we upper-bound P(active cell | i) withP(populated cell | i), which is the probability that cell Ai is pop-ulated by at least one pair of nodes (t, r) in which node t is a po-tential transmitter for a message m in step i, and r is a potentialreceiver of message m.

Note that the occurrence of the event cell Ai is populated isnecessary for the occurrence of the event cell Ai is active.Furthermore observe that P(populated cell | i) coincides withP(active cell | i) in saturated traffic conditions, i.e., when nodesare constantly backlogged with step-i messages to transmit.

The event cell Ai is populated no longer depends on traffic con-ditions, but only on the positions of the nodes in the current step.This fact allows us to design areas Ai using only geometric con-siderations. In particular, when i = imax, t is required to havea home-point in Ai, whereas r is the final destination of messagem, thus it must have a home-point in the same imax-level squareletcontaining Ai. For i < imax, t is required to have a home-point inAi, whereas r must have a home point in the (i+1)-level squareletcontaining the destination.

As we will see, by an appropriate choice of Ai, it is possible toguarantee that limn→∞ P(populated cell| i) > 0 irrespective of thespecific message m in step i stored by t. Moreover we will showthat, with our choice of Ai, the network queues can be loaded issuch a way that P(active cell | i) = Θ(P(populated cell | i)) i.e.,the probability to find a transmitter t holding a message m in step iis Θ(1) 4. This will permit us to conclude that our design of areasAi is indeed optimal (in order sense) also considering queueingeffects.

To compute the minimum value of Ai which guarantees thatlimn→∞ P(populated cell | i) > 0, we separately consider thenumber of candidate transmitters (nodes t) and the number of can-didate receivers (nodes r) within Ai.

Lemma 1. At any given slot, the number of nodes t having one

9 4This condition can be satisfied, more precisely, for the routingstep that determines the overall network capacity, i.e., the systembottleneck.

home-point inAi and currently staying inAi is Θ(Ai) w.h.p., pro-vided that Ai = Ω(log n).

PROOF. see Appendix. A

Lemma 1 allows to say that, to guarantee the presence of at leastone transmitter in Ai, it is sufficient to have Ai = Ω(log n). Theabove lemma is essentially a consequence of concentration resultsfor the distribution of the home-points and of the fact that, whenδ > 2, each node spends a finite fraction of time within a finitedistance from its home-point.

We first consider separately the last step i = imax, in which ttransmits directly to the final destination of the message. Recallthat, by construction, |Simax | = Ω(log n): therefore, by settingAimax = |Simax |, we can be sure that, for any chosen transmittert, the destination can be found in Aimax with non-null probabilityirrespective of the specific message in step i = imax stored by t.

We now consider the generic step i < imax. By lemma 1 the pres-ence of at least one transmitter t in Ai is guaranteed by Ai =Ω(log n). For the generic message in step i stored by t, there areessentially three classes of potential receivers r that can be foundin Ai:

1. nodes b having one home-point placed in the same (i + 1)-level squarelet of the destination, and currently staying inAi

for effect of mobility around such home-point, at distanceΘ(zi).

2. nodes b having one home-point Hhb in the same (i + 1)-

level squarelet of the destination, another home-point H3−hb

placed in Ai, and currently moving within Ai for effect ofmobility around H .

3. nodes b having one home-point Hhb in the same (i+1)-level

squarelet of the destination, another home-point H3−hb not

placed in Ai, and currently moving within Ai for effect ofmobility around H3−h

b .

Notice that classes 2 and 3 require nodes to have multiple home-point, otherwise in the case of a single home-points per node theonly contribution comes from class 1.

We denote by Φji , j = 1, 2, 3, the density of receivers within

Ai belonging to the above three classes, and by Φi =∑3

j=1 Φji

the overall density of receivers within Ai. Then the optimalchoice for Ai is to set Ai = 1/Φi, as proved in [23] (Lemma2). Notice that the overall density of receivers in Ai is equiv-alent (in order sense) to the largest among the Φj

i ’s, henceAi = Θ

((maxΦ1

i , Φ2i , Φ

3i )−1

).

As shown in Appendix B, we have Φ1i = Θ(z2−δ

i ); Φ2i =

Θ(z2−ζi /G(ζ)); Φ3

i = O(max(Φ1, Φ2)). Neglecting the contri-bution of the third class, it remains to understand which one be-tween the first two classes provides the higher density of candidatereceivers. It turns out that this can depend not only on the values ofthe exponents δ and ζ, but also on zi, i.e. on the current step i.

We identify the following three cases:

• Single home-point per node. In this case there is onlyone-class of receivers, whose density is Φ1

i , hence Ai =Θ(zδ−2

i ), ∀i. We observe that Φ1i decreases with zi, hence

the crucial step (the one that requires the largest cells, of areaAi, in order to find candidate receivers) is the first one (cor-responding to i = 0). To guarantee that Ai = O(z2

i ) it mustbe δ ≤ 4. At last, observe that Lemma 1 requires Ai tobe Ω(log n) for any i. Thus zimax must be Ω((log n)

1δ−2 ).

As a result, we select zimax = Θ((log n)1

δ−2 ). Noticethat with this choice we also satisfy the condition zimax =Ω(√

log n), for any δ ≤ 4.

• Multiple home-points per node, ζ > 2. In this case,recalling that we assume ζ < δ so as to have well dis-tinct home-points per node, we have Φ2

i = ω(Φ1i ), hence

Ai = Θ(zζ−2i ), ∀i (we remind that for ζ > 2 we have

G(ζ) = Θ(1)). Again, the first step is the one requiringthe largest value of Ai. To guarantee that Ai = O(z2

i ) itmust be ζ ≤ 4. Similarly to the previous case zimax must beΩ((log n)

1ζ−2 ); thus we select zimax = Θ((log n)

1ζ−2 ).

• Multiple home-points per node, ζ < 2. This is the mostinteresting case, since here Φ1

i decreases with zi, whereasΦ2

i = Θ(z2−ζi /n(2−ζ)/2) increases with zi. The optimal

strategy is to exploit receivers belonging to the second classfor all of the initial steps characterized by zi > zi∗ , wherezi∗ is a critical distance from the destination, and then to relyon receivers belonging to the first class to deliver the messageup to the final destination. This means that the presence ofmultiple home-points is employed to advance the message upto the critical distance zi∗ , after which the mobility arounda single home-point is used. The critical distance zi∗ can beeasily computed by looking for the step i∗ at which Φ1

i =

Θ(Φ2i ). It turns out that zi∗ = Θ(n

2−ζ2(δ−ζ) ). Notice that in

this case the largest cell size among all steps is Ai∗ = zδ−2i∗ .

At last, to guarantee that Ai = O(z2i ) it must be δ ≤ 4. In

this case we select zimax = Θ((log n)1

δ−2 ).

Note that in all cases above we have imax = Θ(log n). The casesin which Ai = O(z2

i ) does not hold require to adopt alternativeschemes that will be described later in Section 6.

Having found the optimal value of Ai, from (4) it turns out thatthe number of parallel transmissions E[Ni] is upper-bounded byNi = n

Ai.

We now turn our attention to the probability distribution psi accord-

ing to which slots are assigned to the different steps of the routingalgorithm. To maximize the system throughput a natural choicewould be to equalize the average number of transmissions that canoccur at each step, and thus avoid that a particular step becomesthe system bottleneck; this is obtained by making ps

i inversely pro-portional to N i. This choice has been pursued in [22], where opti-mal delay-throughput trade-offs have been investigated for the caseof δ < 2. However this approach, when applied to our class ofscheduling-routing schemes specifically tailored to the case δ > 2,results into excessively large delays, because some steps turn out tobe scheduled very rarely.

An alternative strategy, which allows to obtain significant better de-lays at the expense of just a logarithmic penalty in system through-

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

step0

step1

step imax

step0

step1

step imax

exit exit

external

arrival

external

arrival

node a node b

Figure 3: Queue structure of the system. Every node isequipped with one FIFO queue for each step.

put, is to assign an equal probability to each routing step:

psi =

1

imax + 1(5)

We have decided to adopt this slot assignment to obtain the bestthroughput-delay performance by our new class of scheduling-routing schemes. With all these design choices, an upper bound tothe traffic that can be transferred through the network is providedby:

Λ = mini

psi N i =

Θ(n1+ 2−δ

2

log n

) single home-point,2 < δ < 4

Θ(n1+ 2−ζ

2

log n

) multiple home-points,2 < ζ < 4

Θ(n

1+(2−ζ)(2−δ)

2(δ−ζ)

log n

) multiple home-points,ζ < 2 , 2 < δ < 4

(6)We recall that the above expression provides the system throughputin saturated conditions, i.e., when nodes are constantly backloggedby messages to send, at any step.

5.2 Queueing dynamics; maximum through-put

As illustrated in Figure 3, the system can be represented as an opennetwork of queues, where queues correspond to the physical FIFOqueues present in the nodes (recall that in any node there is onequeue for each routing step).

The resulting queueing network is acyclic. Indeed, messages ad-vance in the network visiting queues associated to increasing stepindices i, which guarantees the absence of loops (see Figure 3).

Therefore, considering that traffic is split evenly (for symmetryamong the queues associated to each routing step) among thequeues storing messages in the same routing step, we can be surethat no queue is overloaded by limiting the offered load Λ = nλ toa value strictly less than the saturated system throughput Λ.

We conclude that offered load Λ = O(Λ) can be successfully trans-ferred through the network. In particular, the maximum achiev-able throughput according to our scheme is Λ = Θ(Λ), and inthis case, at least for the index i∗ such that ps

i∗N i∗ is the mini-mum throughput among all steps, we have P(active cell | i∗) =Θ(P(populated cell | i∗).

5.3 Delay analysisTo analyze the delay performance, it is more convenient to lookat the system from a different perspective, in which we associateone virtual queue to each cell, for all steps. A virtual queue storesall messages in a given routing step which have to be transmittedin one of the cells associated to the step. We obtain also in thiscase an open, acyclic queueing network, since messages visit vir-tual queues associated to increasing step indices.

We observe that, in a generic slot, a transmission in a given virtualqueue associated to step i takes place, provided that the consideredqueue is not empty, if the following conditions hold: i) the consid-ered slot is devoted to step i; ii) the corresponding cell belongs tothe subset of non interfering cells that are allowed to transmit; iii)the scheduling policy is able to identify within the correspondingcell a pair of nodes (t, r), which can act respectively as transmitterand receiver of a message enqueued in the considered virtual queue.We model the considered virtual queue as a single server queue, ofwhich we characterize the service time and the arrival process.

We start from the service time. Note that the probability of jointoccurrence of conditions i) and ii) above is ps

i /M , independentlyfrom slot to slot. Slightly more difficult is the evaluation of the oc-currence probability of condition iii), since it depends on the num-ber of messages stored in the considered virtual queue; indeed mes-sages enqueued in the virtual queue can be transmitted only if thenodes physically carrying them are currently staying in the corre-sponding cell. The probability that at least one such transmittersis found within the corresponding cell increases as the number ofmessages stored in the virtual queue increases. However, we canclearly bound (from below) the occurrence probability of iii), as-suming that just a single message is stored in the virtual queue atany time. In this latter case, the probability of finding a (t, r) paircan be easily shown to be Θ(1) as result of our design choices. In-deed, considering a node t having a packet to transmit in step i. Byconstruction it has one home-point Hh

t (h = 1 or 2) within a cellAi of area Ai = Ω(log n). Thus, the probability that node t canbe found in Ai is P(t ∈ Ai|Hh

t ∈ Ai) = Θ(1) as shown in Ap-pendix A. Now, since the average density of potential receivers rinAi is Θ(1/Ai), the average number of potential receivers withinAi is Θ(1). This implies that the probability of finding at least onereceiver within Ai is Θ(1) as well.

Previous arguments permit to conclude that service times of theconsidered virtual queue can be dominated by a sequence of geo-metrically distributed i.i.d. service times with average Θ(1/ps

i ) =Θ(imax).

The arrival process of packets to the virtual queue is, instead, givenby the superposition of the arrival processes to the physical queuesassociated to step i of the nodes having one home-point in the cor-responding cell (the number of such nodes in Θ(Ai)).

For the initial step i = 0, it is just the superposition of the in-dependent Bernoullian processes according to which packets aregenerated at nodes. The analysis of the following steps i > 0 is, in-stead, slightly more complex. Each node b having one home-pointin the corresponding cell can potentially bring a packet to the vir-tual queue in a given slot, if in the considered slot it acts as receiverfor a message in routing step j = i− 1.

This can happen under the following conditions: i) node b moves toa distance Θ(zi) from its home-point; ii) the slot is assigned to step

j = i− 1; iii) the visited cell has a packet to transmit; iv) the nodeis chosen as receiver among the candidate receivers in the same cellbelonging to step j.

The satisfaction of the above conditions (especially condition iii))can not be easily evaluated. However, observe that, provided thatthe slot is assigned to step j = i − 1, by construction the numberof simultaneous arrivals at the considered virtual queue is upper-bounded by the number of nodes having an home-point in Ai thatare concurrently staying at a distance Θ(zi) from Ai.

The number of such nodes is i) binomially distributed since posi-tions of different nodes are i.i.d. (it tends to a Poisson distributionin the large n limit); ii) it regenerates from slot to slot. iii) its aver-age is Θ(1) as shown in Appendix C.

As a result of previous arguments, although the dynamics of a vir-tual queue associated to step i may exhibit complex dependen-cies over time, the number of messages in it can be stochasti-cally dominated by the number of messages in a Geo[X]/Geo/1 (aGeo/Geo/1 queue with bulk arrivals), in which the service times(batch sizes) are memoryless and lower (upper)-bounded as ex-plained above.

Thus, the average delay E[Di] experienced by step-i messages inthe considered virtual queue can be bounded using the Pollacek-Kinchine formula [24]:

E[Di] ≤ si +a2s2

i − a(s(2)i + si) + a(2)s2

i

2(1− asi)

where si and s(2)i are, respectively, the first and the second moment

of the service time for the dominating Geo[X]/Geo/1 queue, whilea and a(2) are, respectively, the first and the second moment of thebatch size distribution (which is binomial both in the case i = 0and in the case i > 0).

From the above considerations on service times and batch sizedistributions it follows that: si = Θ(imax), s

(2)i = Θ(i2max),

ai = Θ(1/imax) for every i, while a(2)i is Θ(1/i2max) for i = 0

and Θ(1/imax) for i > 0.

In conclusion, recalling that imax = Θ(log n), it turns out thatE[Di] = Θ(log n) for every step i.

At last, since messages are forwarded to destinations throughO(imax) = O(log n) steps, the overall delivery delay is E[D] =O((log n)2).

6. ALTERNATIVE SCHEMES FORLARGE VALUES OF δ AND ζ

We recall that our previously described scheduling-routing schemesrequire that Ai = R2

i = O(z2i ). This condition does not hold when

δ > 4, or when ζ > 4 (in the case of multiple home-points).

In the case of multiple home-points having ζ < 2 and δ > 4, it isstill possible to apply our bisection scheme up to a minimum dis-tance z∗ from the destination, and then resort to a direct transmis-sion to reach the destination. Indeed, no matter how big is δ, i.e.,how much restrictive is the mobility of nodes around each homepoint, we can still exploit the presence of multiple home-point pernode to efficiently advance the message in the beginning of theroute, when zi > z∗. The value of z∗ can be computed consid-

ering that the density of receivers is at least Φ2i = Θ(z2−ζ

i /G(ζ)),

and thus condition Ai = R2i = O(z2

i ) provides z∗ = Θ(n2−ζ

2(4−ζ) ).When the message has reached distance z∗ from the destination,it can be sent directly to the destination by a single hop which in-troduces a delay of Θ(1), since every node spends a constant frac-tion of time in proximity of its home-point. The above alternativeschemes provides a throughput

Λ = Θ(n

1− 2−ζ4−ζ

log n

)multiple home-points, ζ < 2 , δ > 4 (7)

and delay D = O((log n)2). It can be observed that this is thesame performance achieved by our original scheme when δ = 4.

In the case of a single-home point per node with δ > 4, or multiplehome-point per node with ζ > 4, instead, our bisection schemecannot be applied even in the first steps. In this cases it is necessaryto resort to an alternative scheme valid for any value of δ > 2,that does not exploit neither the multiplicity of home-points nor themobility of nodes.

This scheme is based on the observation that, for any δ > 2, everynode spends a constant fraction of time within a finite distance fromits home-point. Hence, by enabling the nodes to communicate onlywhen they are in proximity of their home-points, we can apply theclassic Gupta-Kumar scheme for fixed nodes at thus achieve at leastthe same performance as that of a static network.

Recall from [1, 8] that a static network with uniform node den-sity sustains a per-node throughput λ = Θ(1/

√n log n), while in-

curring a delay D = Θ(√

n/ log n). More in general, it allowsto achieve delay-throughput trade-offs characterized by λ/D =Θ(1/n), for all λ = O(1/

√n log n) [8].

7. SUMMARY OF RESULTSTo summarize our results, we report the scaling exponent5 ofthe power function λ/D achievable by our family of scheduling-routing schemes, in all considered cases:

e(λ/D) =

2−δ2

single home-point, δ ≤ 42−ζ2

multiple home-points, 2 < ζ ≤ 4

−1single home-point, δ > 4

or multiple home-points, ζ > 4

(2−ζ)(2−δ)2(δ−ζ)

multiple home-points, ζ < 2 , δ ≤ 42−ζζ−4

multiple home-points, ζ < 2 , δ > 4

(8)Figure 4 offers a 3D representation of (8). In Figure 4, points char-acterized by ζ > δ represent nodes having a single home-point.

In Figure 5 we report a 2D plot of the same results restricting ourattention to cases in which ζ < 2, or in which nodes have a singlehome-point. In particular, the curve labelled ‘1 h.p.’ correspondsto the novel scheduling-routing scheme proposed in this work forthe single home-point case, whereas the curve labelled ‘1 h.p. oldscheme’ refers to the performance achievable in the same case bythe scheme proposed in [22], characterized by e(λ/D) = 2 − δ.Figure 5 shows that, when δ > 2, the scheme proposed in this work

9 5The scaling exponent of a generic function f(n) is defined as

e(f) := limn→∞log f(n)

log n.

2 3

4 5

6 7

8 0 1 2 3 4 5 6

-1

-0.8

-0.6

-0.4

-0.2

0

e (λ/D)

δ

ζ

Figure 4: Scaling exponent of the power achievable by the pro-posed schemes for different values of ζ and δ.

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

2 3 4 5 6 7

e (λ

/D)

δ

ζ = 0ζ = 0.5

ζ = 1ζ = 1.5

1 h.p.1 h.p. old scheme

Figure 5: Scaling exponent of the power achievable by the pro-posed schemes as a function of δ, for different values of ζ < 2(multiple home-points) and for the single home-point case.

outperforms the one proposed in [22].

7.1 Optimality of the proposed schemesIn this section we show that our schemes jointly achieve near-optimal throughput and delay performance, at least over a signif-icant range of parameter values.

We start recalling two results that have been proved in [23].

Lemma 2. In a scenario in which nodes move around a singlehome-point, no scheduling-routing scheme can achieve a through-put larger than (in order sense):

λ =

Θ(1) δ ≤ 2

Θ(n2−δ2 (log n)

1−δ2 ) 2 ≤ δ < 3

Θ(n1/2) δ ≥ 3

Lemma 3. In a scenario in which nodes move around twohome-points, under the assumption (ζ < δ) no scheduling-routing

scheme can achieve a throughput that is in order sense larger than:

λ =

Θ(1/ log n) ζ < 2

Θ(n2−ζ2 ) 2 ≤ ζ < 3

Θ(n1/2) ζ ≥ 3

Comparing the results obtained by our schemes with the above up-per bounds on the throughput, and observing that by constructionD = Ω(1) under any possible scheme, we obtain:

Theorem 1. The proposed schemes jointly achieve optimalthroughput and delay exponents for 2 < δ ≤ 3, in the case ofsingle home-point, and for 2 < ζ ≤ 3, in the case of multiplehome-points.

8. CONCLUSIONSPrevious work on asymptotic delay-throughput performance in mo-bile ad hoc networks has mostly been done under the assumptionthat the stationary distribution of each node is uniform over the net-work area. In [22] we have shown that this condition can be largelysuboptimal. Indeed delay-throughput performance can be dramati-cally improved when each node moves around one home-point ran-domly placed in the area, according to a restricted mobility processwhich results into a spatial stationary distribution that decays as apower law of exponent δ with the distance from the home-point.

In this paper we have introduced a new class of scheduling androuting schemes that significantly outperform the schemes in [22]for δ > 2. In particular, the schemes introduced in this paperjointly achieve optimal throughput and delay scaling exponents for2 < δ ≤ 3. We have also extended our analysis to the case of mul-tiple home-points per node, showing that the network performancecan be further improved by home-point multiplicity. In this casethe proposed scheduling-routing schemes jointly achieve optimalthroughput and delay scaling exponents for 2 < ζ ≤ 3.

9. REFERENCES[1] P. Gupta, P.R. Kumar, “The capacity of wireless networks”,

IEEE Trans. on Information Theory, 46(2), pp. 388–404,2000.

[2] M. Grossglauser, D.N.C. Tse, “Mobility increases thecapacity of ad hoc wireless networks”, IEEE/ACM Trans. onNetworking, vol. 10, no. 2., August 2002.

[3] Delay Tolerant Network Research Group:www.dtnrg.org

[4] A. Chaintreau, P. Hui, J. Scott, R. Gass, J. Crowcroft, and C.Diot, “Impact of human mobility on opportunisticforwarding algorithms," IEEE Transactions on MobileComputing, 6(6): 606–620, June 2007.

[5] J. Burgess, B. Gallagher, D. Jensen, B. N. Levine, “MaxProp:Routing for Vehicle-Based Disruption-Tolerant Networking",in Proc. IEEE INFOCOM, Barcelona, Spain, April 2006

[6] M. J. Neely, E. Modiano “Capacity and delay trade offs forad-hoc mobile networks” IEEE Trans. on InformationTheory, 51(6), pp. 1917–1937, 2005.

[7] L. Ying, S. Yang, R. Srikant, “Optimal Delay-ThroughputTradeoffs in Mobile Ad Hoc Networks," IEEE Trans. onInformation Theory, 54(9), pp. 4119–4143, 2008.

[8] A. El Gamal, J. Mammen, B. Prabhakar, and D. Shah,“Throughput-Delay Trade-off in Wireless Networks", inProc. IEEE INFOCOM ’04.

[9] N. Bansal, Z. Liu, “Capacity, Delay and Mobility in WirelessAd-Hoc Networks," in Proc. IEEE INFOCOM ’03.

[10] G. Sharma, R. R. Mazumdar and N. B. Shroff, “Delay andCapacity Trade-offs in Mobile Ad Hoc Networks: A GlobalPerspective" in Proc. IEEE INFOCOM ’06.

[11] J. H. Kang, W. Welbourne, B. Stewart, G. Borriello,“Extracting Places from Traces of Locations", ACM MobileComputing and Communications Review, 9(3), July 2005.

[12] M. Balazinska, P. Castro, “Characterizing Mobility andNetwork Usage in a Corporate Wireless Local-AreaNetwork," in Proc. MobiSys ’03.

[13] N. Sarafijanovic-Djukic, M. Piorkowski, and M.Grossglauser, “Island Hopping: Efficient Mobility-AssistedForwarding in Partitioned Networks", IEEE SECON 2006.

[14] J. Leguay, T. Friedman, V. Conan, “Evaluating MobilityPattern Space Routing for DTNs", in Proc. IEEE INFOCOM’06.

[15] I. Rhee, M. Shin, K. Lee, S. Chong. S. Hong, “On theLevy-walk Nature of Human Mobility," in Proc. INFOCOM’08

[16] S.N. Diggavi, M. Grossglauser, D.N.C. Tse, “Evenone-dimensional mobility increases ad hoc wirelesscapacity”, IEEE Trans. on Information Theory, 51(11), pp.3947–3954, 2005.

[17] J. Mammen and D. Shah, “Throughput and Delay inRandom Wireless Networks with Restricted Mobility," IEEETrans. on Information Theory, 53(3), pp. 1108–1116, 2007.

[18] M. Garetto, P. Giaccone, E. Leonardi, “Capacity Scaling inDelay Tolerant Networks with Heterogeneous MobileNodes" in Proc. ACM MobiHoc ’07.

[19] M. Garetto, P. Giaccone, E. Leonardi, “Capacity Scaling ofSparse Mobile Ad Hoc Networks”, in Proc. INFOCOM ’08.

[20] F. Xue, P. R. Kumar, “Scaling laws for ad hoc wirelessnetworks: an information theoretic approach", Found. TrendsNetw., vol. 1, no. 2, pp. 145–270, 2006.

[21] L. Kleinrock, "On Flow Control in Computer Networks", inProc. ICC ’78.

[22] M. Garetto and E. Leonardi, “Restricted Mobility ImprovesDelay-Throughput Trade-offs in Mobile Ad-Hoc Networks,"submitted for publication, available athttp://arxiv.org/abs/0807.1228

[23] M. Garetto, P. Giaccone, E. Leonardi, “Capacity Scaling inAd Hoc Networks with Heterogeneous Mobile Nodes: theSub-critical Regime”, accepted for publication on ACM/IEETransactions on Networking available on-line at:http://www.telematica.polito.it/leonardi/papers/TON_sub.pdf

[24] H. Takagi, Queueing Analysis : Discrete-Time Systems,North Holland, 1993.

APPENDIXA. PROOF OF LEMMA 1Consider one node a having a home point Hh

a (h = 1 or 2), withina cell Ai of area Ai = Ω(log n). The probability that node a isfound in Ai can be evaluated as

P(a ∈ Ai|Hha ∈ Ai) > pHh

a

∫

Ai

φ(||X −Hha ||) dX = Θ(1)

being pHha

= Θ(1) the probability that node a is associated to itsh-th home-point. The integral in the above expression is Θ(1) forany δ > 2. Now, let NH∈Ai be the number of nodes having onehome-point inAi. The average number of nodes having one home-

point in Ai, that can be found in Ai is, by definition:

E[NAi ] = NH∈AiP(a ∈ Ai|Hha ∈ Ai) > Θ(NH∈Ai)

Since the home-points are Poisson distributed, whenever Ai =Ω(log n), the number of nodes NH∈Ai having a home-point in Ai

is w.h.p. Θ(Ai), and we get the assert.

B. DIMENSIONING OF Ai FOR i < imax

From Lemma 1 it directly descends that the average density oftransmitters within Ai is Θ(1) w.h.p. whenever Ai = Ω(log n),as it is given by E[NAi ]/Ai.

Now we turn our attention to receivers b and without lack of gener-ality, we assume

√2Ai < εzi for some ε << 1.

We remind that there are essentially three classes of potential(i + 1)-relays b that can be found in Ai: i) nodes b having onehome-point placed in the same (i + 1)-level squarelet of the des-tination, and currently staying in Ai for effect of mobility aroundsuch home-point, at distance Θ(zi). ii) nodes b having one home-point Hh

b in the same (i + 1)-level squarelet of the destination, an-other home-point H3−h

b placed inAi, and currently moving withinAi for effect of mobility around H3−h

b . iii) nodes b having onehome-point Hh

b in the same (i + 1)-level squarelet of the desti-nation, another home-point H3−h

b not placed in Ai, and currentlymoving within Ai for effect of mobility around H3−h

b .

Let us consider a node moving around a home-point placed in thesame squarelet Sk

i+1 of the destination; the probability that thisnode is found in Ai is;

P(b ∈ Ai|Hhb ∈ Sk

i+1) = pHhb

∫Ai

φ(||X −Hhb ||) dX

= Θ(Aiz−δi )

Thus, considering that the number of nodes having a home-point inSk

i+1 is Θ(z2i ), the average density of such nodes in Ai is Φ1

i =

z2−δi .

Now we evaluate the probability that a node b having one home-point Hh

b (with h = 1 or 2) in Ski+1 has the other home point H3−h

b

in a cell Ai located at distance zi from Ski+1:

P(H3−hb ∈ Ai | Hh

b ∈ Ski+1) =

∫Ai

ψ(||X −Hhb ||) dX

= Θ(

Aiz−ζi

G(ζ)

)

Now, considering that the number of nodes having one home-pointin Sk

i+1 is Θ(z2i ) and that nodes having an home-point in Ai have

a non-null probability to be found in Ai, we easily get that Φ2i =

z2−ζi

G(ζ).

At last we evaluate the probability that a node b having one home-point Hh

b (with h = 1 or 2) in Ski+1, and the other home-point

H3−hb outside Ai, can be found in Ai.

P(b ∈ Ai, H3−ha ∈/Ai|Hh

b ∈ Ski+1) =

= pH3−h

b

∫Ai

∫O\Ai

φ(||X −H3−hb ||)ψ(||H3−h

b −Hhb ||)

dH3−hb dX

(9)Defining as BX,ε the R2-ball of center X and radius εzi, we splitthe inner integral into the sum of three contributions. The first iscomputed over an area BHh

b,ε: in this case the term ψ(||H3−h

b −

Hhb ||) can be considered Θ(1). Taking into account the triangular

inequality ||X−H3−hb ||+||H3−h

b −Hhb || ≥ ||X−Hh

b || = Θ(zi),the first contribution becomes:

∫Ai

∫B

Hhb

,ε

φ(||X −H3−hb ||) dH3−h

b dX

≤ ∫Ai

∫B

Hhb

,ε

φ(||X −Hhb ||) dH3−h

b dX

= O(Φ1i )

The second contribution is computed on BX,ε \Ai: using the sameconsiderations above, we can say that it is O(Φ2

i ). At last, the thirdcontribution computed overO\(BHb,ε∪BX,ε) can be easily shownto be negligible. Indeed, note that uniformly over the considered

domain we have: ε1+ε

≤ ||X−H3−hb

|||H3−h

b−Hh

b|| ≤

1+εε

.

Thus, being function ψ() non increasing we have:∫Ai

∫O\(BHb,ε∪BX,ε)

φ(||X −H3−hb ||)

ψ(||H3−hb −Hh

b ||) dH3−hb dX

≤ ∫Ai

∫O\(BHb,ε∪BX,ε)

φ(||X −H3−hb ||)

ψ(

ε1+ε

||X −H3−hb ||) dH3−h

b dX

= Θ(Aiz2−(δ+ζ)i )

(10)

Combining previous results we can conclude that Φ3i =

O(max(Φ1i , Φ

2i )).

C. EVALUATION OF THE NUMBER OFNODES AT DISTANCE Θ(zi) FROMTHE HOME-POINT

For the sake of the brevity we restrict to the case of a single homepoint. The generalization to the multiple home-point case can beeasily carried out following the same rationale as in Appendix B.

Let us consider a node a having a home-point in Ai. The probabil-ity that this node is found at a distance Θ(zi) from its home-pointis:

P(zi) = 2π

∫ Czi

czi

ρφ(ρ) dρ = Θ(z2−δi )

being 0 < c < C < ∞ two arbitrary constants. Now, since thenumber NH∈Ai of nodes having one home-point in Ai is, w.h.p,Θ(Ai) and being Ai = zδ−2

i , it follows that the average numberof nodes having one home point in Ai that are found at distanceΘ(zi) from their home-point is P(zi)NH∈Ai = Θ(1).